[/quote]
,
,such that 
P
be a sequence of positive real numbers such that for every 
,![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
,![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
,
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
,
and non-negative reals 
,
Here 
and 
denote the median and arithmetic mean of 
,
,
an arbitrary point on side 
.
meets the circumcircle of 
again at 
,
meets it again at 
.
,
and 
are concurrent.
be non-negative real numbers. Prove that
be a triangle such that 
.
lie on segments 
,
and 
are the angle bisectors of triangle 
.
.
satisfy 
prove in equality.
and circumcenter 
.
with center 
intersects the segment 
and 
,
,
are all different and lie on line 
be the points of intersection of 
be the second point of intersection of the circumcircle of triangle 
.
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
.
.
be a point inside a triangle 
such that![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](http://latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
,
be the incenters of triangles 
,
,
,
,
meet at a point.
satisfying
Prove
cups labeled 
,
-
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
for which 
divides 
.
Y by
Let 
be a triangle with 
, 
, 
. Let 
, and 
be points lying on sides 
, respectively. Given that 
, and 
are cyclic respectively, and the circumcircle of 
are tangent to line 
, find the length of segment 
.
be a triangle with 
, 
, 
. Let 
, and 
be points lying on sides 
, respectively. Given that 
, and 
are cyclic respectively, and the circumcircle of 
are tangent to line 
, find the length of segment 
.This post has been edited 3 times. Last edited by NicoN9, Apr 20, 2025, 11:22 AM
Reason: typo fixed
Reason: typo fixed
